Regression Models Using Shapes of Functions as Predictors
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Functional variables are often used as predictors in regression problems. A commonly-used parametric approach, called scalar-on-function regression, adopts the standard inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability because phase changes can have disproportionately large influence on the response variable. A simple solution is to perform phase-amplitude separation (as a pre-processing step) and then apply functional regression model. In this paper, we propose a different approach, termed elastic functional regression, where phase separation is performed inside the regression model, rather than as pre-processing. This approach involves multiple notions of phase and is based on the Fisher-Rao metric instead of the standard L2 metric. Due to its superior invariance properties, this metric allows more immunity to phase components and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals and the historical stock market.
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